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Contract No. DE-AC36-99-GO10337 

D.J. Friedman, A.J. Ptak, Sarah R. Kurtz, 

and J.F. Geisz 

Prepared for the 31st IEEE Photovoltaics Specialists 

Conference and Exhibition 

Lake Buena Vista, Florida 

January 3–7, 2005 

February 2005 • NREL/CP-520-37418 

Analysis of Depletion-Region 

Collection in GaInNAs Solar 

Cells 
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Analysis Of Depletion-Region Collection In GaInNAs Solar Cells 

D.J. Friedman, A.J. Ptak, Sarah R. Kurtz, and J.F. Geisz 

National Renewable Energy Laboratory, Golden, CO 80401 USA 

ABSTRACT 

We provide qualitative insight into depletion-region 

collection in GaInNAs cells to (1) understand the effect of 

diffusion length L on the QE; and (2) describe the 

magnitude of L required to get adequate current from the 

cell. We use Wolf’s equations for the QE including a drift 

field E, and model E as being equal to the junction built-in 

voltage distributed uniformly across the depletion region. 

This allows us to calculate the QE as a function of L and 

depletion width WD. We show that if L is sufficiently small, 

increasing WD can actually decrease the QE. To determine 

how long L needs to be in a practical GaInNAs junction, 

we calculate from the QE the short-circuit current density

as a function of WD and L. This allows us to estimate that 

Lambipolar needs to be greater than roughly 1 μm in order to 

obtain enough photocurrent for the 4-junction application, 

giving guidance to the experimental effort to develop such 

cells. 

INTRODUCTION 

Considerable effort has been spent on developing 1-

eV GaInNAs junctions for future-generations of highefficiency multijunction devices [1-5]. However, the poor 

diffusion lengths in this material result in unacceptably low

quantum efficiencies (QE): as-grown pn junctions typically

show QE < 20%. 

With the introduction of a wide depletion region, i.e. a 

pin structure, to collect photocarriers, greatly improved 

QEs > 70% have been achieved [2]. Considerable 

progress has been made in five- and six-junction 

structures for which this magnitude of QE is sufficient, due 

to the reduced current-density requirements on the 

GaInNAs junction for current-matching in these structures 

[4,5]. However, for application in the simpler 

GaInP/GaAs/GaInNAs or GaInP/GaAs/GaInNAs/Ge 

structures, QE>90% is required. Such QEs have not been 

achieved for 1-eV GaInNAs grown by MOVPE, although 

significant progress has recently made using MBE growth 

[6]. One problem has been the difficulty of obtaining 

sufficiently large depletion widths WD. However, merely

obtaining large WD is not sufficient. 

The drift field in the depletion region helps move the 

photocarriers to the junction, resulting in an effective 

collection length LC for photocarriers which is enhanced 

compared to the (zero-field) diffusion length L. It is 

conventionally assumed that every photocarrier in the 

junction depletion region is swept to the junction by the 

drift field and collected; i.e. that LC > WD. For this to be the 

case, it is necessary for the photocarriers to live long 

enough for the field to sweep them to the junction. 

However, if the diffusion length L is very low, even the 

enhancement in LC provided by the drift field may not be 

enough to get the required LC > WD. 

The consequence would be a QE lower than if there 

were complete collection of carriers from the depletion 

region. Figure 1 gives an experimentally observed 

example. It shows the measured QE for a GaInNAs 

junction whose WD = 0.3 μm as determined by

capacitance-voltage profiling. For comparison, the dashed 

line is what the QE would be for complete collection in the 

depletion region, showing that in this cell a significant 

fraction of the photocarriers generated in the depletion 

region are not being collected. In contrast, if we get rid of 

the oversimplified assumption of complete collection in the 

depletion region, we can account for the measured QE as 

shown by the solid line, to be discussed in more detail 

below. 

Fig. 1. Dots: measured QE of a GaInNAs junction with 

WD=0.3 μm. Dashed line: modeled QE assuming complete 

collection in the 0.3 μm depletion region. Solid line: 

Modeled QE assuming incomplete collection (see text). 

Here we focus exclusively on collection in the depletion 

region. The collection of photocarriers in the “flat-band” 

regions of GaInNAs solar cells is discussed by Kurtz [7]. 

OBJECTIVE 

Because the minority-carrier diffusion lengths in 

GaInNAs are low, in order to understand collection in the 

1
depletion region well enough to take advantage of the driftfield enhancement, it is necessary to account for the 

dependence of the collection length LC on the diffusion 

length and on the drift field. Our objective here is to 

provide this understanding by modeling the effect of the 

enhanced collection in GaInNAs junctions, so that we can 

(1) explain the observed phenomenon (illustrated 

qualitatively in Fig. 1) that increasing WD sometimes 

results in decreasing, rather than increasing, QE; and (2) 

understand and describe what magnitude is required for L 

in order to get the required current from the cell. Our 

approach is not intended to be a fully quantitative 

description of the junction, the detailed complexities of 

which (e.g. ambipolar transport, trapping by localized 

states, etc) would be difficult to treat analytically. 

APPROACH 

In order to gain insight into the characteristics of drift 

collection in these devices, we use an idealized model of 

the junction shown in Fig. 2 to capture the essential 

features. The junction is assumed to be fully depleted, with 

a uniform field E=Vbi/WD. 

hν

WD

junction

back

surface

(S=0)

Fig. 2. Schematic of the model for the junction. 

Wolf [8] provides the quantum efficiency of the 

illuminated junction including the field, as: 

 QE = α

G ⋅ 1

b2 −1

⋅ b −1+ (1− a)e−Gd + (a − b)e−bGd

coshGd + a sinhGd

⎡

⎣

⎢

⎤

⎦

⎥ (1) 

where 

 F ≡ eE / 2kT , (2a)

b ≡ (α − F) /G , (2b) 

G ≡ F 2 +1/L2 , (2c)

a ≡ F

G

+

S

GD , (2d) 

α=α(λ) is the absorption coefficient, d=WD is the base 

thickness, S is the back surface recombination velocity, 

D= kBTμ/e is the diffusion coefficient at temperature T, and 

L=(kBTμτ/e)1/2 is the diffusion length. The carrier mobility μ 

and lifetime τ are assumed to be independent of field. We

take Vbi=0.75V for a 1-eV-bandgap junction, and for 

simplicity assume S=0. These equations then allow us to 

calculate the QE as a function of WD and L. 

RESULTS 

Using our model, we can now derive insight into the 

effects of diffusion length and field (or cell thickness) in 

determining the QE. Figure 3 compares the QE calculated 

for three different combinations of L and WD. For L=1 μm 

and WD=1 μm, most of the carriers are collected. 

Increasing WD to 3 μm increases the QE because the cell 

is thicker. However, the situation changes qualitatively

when we now consider a much lower diffusion length. At 

L=0.1 μm and WD=1 μm, the carrier collection is much less 

than at L=1μm; i.e., the field is no longer adequate to 

sweep all the carriers to the junction before they

recombine. If we now increase WD to 3 μm, the QE 

actually decreases, rather than increasing as was the case 

for the larger L=1 μm. This is because for small L, the 

detrimental effect of lowering the field (by a factor of 3 in 

this example) outweighs the effect of the increased 

thickness of the region where light is being absorbed. 

Thus, the QE can either increase or decrease with 

increasing WD, depending on the magnitude of L. This is 

the central theme of this paper, taking us beyond the 

simple approximation in which increased WD necessarily

results in enhanced QE. 

Fig. 3. QEs calculated for several combinations of WD and 

L, for the fully depleted structure of Fig. 2. 

We can now go back to the data of Fig. 1 and, given 

the measured WD of 0.3 μm in that device, see what value 

of L gives a QE consistent with the measured data. The 

figure shows that L=0.04 μm is consistent with the 

measured QE. 

Clearly, for practical devices, we cannot permit L to be 

arbitrarily small. To quantify this, we calculate from the QE 

Fig. 4. Contours of JSC as a function of WD and L for the 

low-AOD spectrum, for the model junction of Fig. 2. In the 

grey region, increasing WD decreases JSC. Note that L is 

the diffusion length, not the collection length LC. 

2
3.0

2.5

2.0

1.5

1.0

WD (μm)

0.5 1.0 1.5

diffusion length L (μm)

 Eg=1.0 eV

 Eg=0.95 eV

JSC=15.8 mA/cm2 J 2

Figure 5. Contours of JSC calculated as for Fig. 4, but for 

Eg=0.95 eV as well as Eg=1.0 eV. 

the short-circuit current density JSC as a function of WD

and L necessary to obtain the current needed in the 

multijunction application. Figure 4 shows JSC(L,WD) for the 

low-AOD spectrum. To illustrate the effect of L on whether 

increasing WD is useful, consider two cases. (1) As shown 

by the arrow from A to B in Fig. 4, we set L=0.2 μm, and 

increase WD from 0.3 μm to 0.5 μm. In this case, 

increasing WD increases JSC. (2) As shown by the arrow

from C to D in Fig. 4, we set L=0.1 μm, and increase WD

from 0.5 μm to 2 μm. In this case, increasing WD

decreases JSC. 

The second case is an example of a general feature of 

JSC: there is a region of WD and L, shaded in grey in the 

figure, in which increasing WD decreases JSC. To

understand why this region exists, we need to understand 

in more detail the effect of the drift field. The drift field 

results in a collection length LC =1/(G-F), which is always

greater than or equal to L; i.e. the field enhances the 

collection length. For zero field, LC becomes LC=L, while in 

the limit of high field LC approaches LC=μτE. As long as LC

> WD, all the photocurrent generated in the depletion 

region is collected. However, if LC is reduced so that LC < 

WD, collection in the depletion region is no longer 

complete. A key point in our analysis is that the drift field 

decreases with increasing WD because the same voltage 

is distributed over a longer distance; and hence LC

decreases as well. As long as LC > WD, this will not have a 

significant effect on the photocurrent collection, which is 

determined by the smaller of LC and WD. But, under 

conditions where increasing WD reduces LC to less than 

WD, the current collection will decrease rather than 

increase. The crossover from WD-limited to LC-limited JSC

is given by the boundary of the grey region in Fig. 4. 

For current-matching the 1.0-eV GaInNAs junction 

under GaInP/GaAs junctions, for the low-AOD spectrum a 

JSC of 15.8 mA/cm2

 is needed for ideal top and middle 

cells. The JSC=15.8 mA/cm2

 contour is indicated in bold in 

Fig. 4, and defines the minimum values of WD and L 

required to achieve this current for a fully depleted cell. 

The contour shows that not only do we need WD>2 μm; we

also need L > 1 μm. In contrast, current matching for the 

six-junction structure [4,5] is about 8 mA/cm2

. The 

corresponding contour in Fig. 4 shows that for this current, 

the requirements on WD and L are greatly reduced to 

roughly WD > 0.5 μm and L > 0.2 μm. 

For application in a GaInP/GaAs/GaInNAs threejunction structure, it may be worth considering GaInNAs 

band gaps below 1 eV as a strategy to increasing the 

GaInNAs current. Figure 5 compares the Eg=1 eV

contours of JSC from Fig. 4 with JSC contours calculated for 

Eg=0.95 eV. Reducing Eg from 1 to 0.95 eV reduces the 

values needed for both WD and L, making this appear to 

be a promising approach. Unfortunately, other studies [9] 

indicate that the minority-carrier properties of GaInNAs 

degrade so rapidly with decreasing Eg that achieving L=0.5 

μm at Eg=0.95 eV is probably even more difficult than 

achieving L=1 μm at Eg=1 eV. 

Furthermore, obtaining large WD in practice also 

appears to be increasingly difficult with decreasing band 

gap. Figure 6 plots the required depletion width WD

(assuming LC>WD) as a function of junction band gap Eg

for current-matching under GaInP/GaAs junctions. The 

figure also shows the depletion widths we have obtained to 

date on MOVPE-grown GaInNAs junctions, measured by

capacitance-voltage profiling. At Eg=1 eV, the achieved 

depletion widths are within a factor of three of the needed 

depletion width. However, the achieved depletion widths 

fall off rapidly with decreasing band gap, presumably due 

to an increase in the concentration of either doping 

impurities or defects with increasing nitrogen concentration 

(i.e. decreasing band gap). The achieved depletion widths 

shown in Fig. 2 decrease more rapidly with Eg than the 

needed depletion width decreases, weakening the 

motivation for going to bandgaps below 1 eV. 

However, it may still be possible to achieve the 

needed depletion widths by growing the junctions with 

solid-source molecular-beam epitaxy (MBE) [6]. Using our 

model, we can estimate the diffusion length in the MBEgrown long-depletion-width junctions of Ptak [6]. Figure 7 

compares the QE measured for one of these MBE devices 

(the W=2.7 μm device of Ref. [6]’s Fig. 2) with a family of 

QE curves generated by our model as a function of 

diffusion length. For the absorption coefficient for this 

Eg=1.28 eV device, we use the GaAs absorption coefficent 

Figure 6. Depletion width needed to match the GaInNAs 

third junction’s current to the GaInP and GaAs top two 

junctions in a GaInP/GaAs/GaInNAs three-junction cell. 

Also shown (dots) are the depletion widths that have 

actually been achieved for MOVPE-grown GaInNAs 

junctions to date. 

3
translated down in energy to match the 1.28-eV band 

edge, as discussed by Kurtz [7]. The results of Fig. 7 

suggest that the diffusion length in the base of this device 

is at least 1–2 μm. The next development step would be to 

achieve similar L and WD in a Eg=1.0-eV device. 

SUMMARY 

We have modeled drift collection in idealized fully

depleted 1-eV GaInNAs solar cells to provide insight into 

the dependence of the QE on the diffusion length L and 

depletion width WD. We find that, roughly, L>1 μm as well 

as WD > 2 μm are required for current-matching in the 3- 

or 4-junction application. The Eg=1.28-eV MBE-grown 

device of Fig. 7 [6] appears to satisfy these criteria, 

although not yet with a 1.0-eV band gap. Lowering the 

band gap from 1.0 to 0.95 eV lowers the requirements on 

WD and L, although in practice it may be harder to obtain 

the required WD and L at Eg= 0.95 eV than at 1 eV. 

Alternatively, for the six-junction structure [4,5], a much 

less demanding L>0.2μm and WD>0.5 μm are required, an 

encouraging result for this device structure as these 

paramaters should be obtainable with MOVPE-grown 

material. 

Figure 7. Measured QE for MBE-grown device EB252n4, 

compared to calculated QE curves from our model with 

various diffusion lengths. 

REFERENCES 

[1] S.R. Kurtz, D. Myers, and J.M. Olson, “Projected 

Performance of Three- and Four-Junction Devices 

using GaAs and GaInP”, 26th IEEE Photovoltaic 

Specialists Conference, 1997, pp. 875-878. 

[2] D.J. Friedman, J.F. Geisz, S.R. Kurtz, and J.M. Olson, 

“1-eV solar cells with GaInNAs active layer”, J. Cryst. 

Growth 195, 1998, pp. 409-415. 

[3] J.F. Geisz and D.J. Friedman, “III-N-V semiconductors 

for solar photovoltaic applications”, Semicon. Sci. 

Technol. 17, 2002, pp. 769-777. 

[4] R.R. King et al., “Metamorphic III-V materials, 

sublattice disorder, and multijunction solar cell 

approaches with over 37% efficiency”, 19th European 

Photovoltaic Solar Energy Conference and Exhibition, 

2004, 

[5] M. Meusel et al., “European roadmap for the 

development of III-V multi-junction space solar cells”, 

19th European Photovoltaic Solar Energy Conference, 

2004, 

[6] A.J. Ptak, D.J. Friedman, C. Kramer, and M. Young, 

“Enhanced-depletion-width GaInNAs solar cells grown 

by molecular-beam epitaxy”, 31st IEEE Photovoltaic 

Specialists Conference, 2005, 

[7] S. Kurtz et al., “Effect of nitrogen concentration on the 

performance of GaxIn1-xNyAs1-y solar cells”, 

Proceedings of the IEEE Photovoltaic Specialists 

Conference, 2005, 

[8] M. Wolf, “Drift fields in photovoltaic solar energy

converter cells”, Proc. IEEE 51, 1963, pp. 674-693. 

[9] D.J. Friedman, J.F. Geisz, W.K. Metzger, and S.W.

Johnston, “Trap-dominated minority-carrier 

recombination in GaInNAs pn junctions”, Appl. Phys. 

Lett. 83, 2003, pp. 698-700. 

4
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Analysis of Depletion-Region Collection in GaInNAs Solar Cells 

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14. ABSTRACT (Maximum 200 Words)

We provide qualitative insight into depletion-region collection in GaInNAs cells to (1) understand the effect of diffusion 

length L on the QE; and (2) describe the magnitude of L required to get adequate current from the cell. We use 

Wolf’s equations for the QE including a drift field E, and model E as being equal to the junction built-in voltage 

distributed uniformly across the depletion region. This allows us to calculate the QE as a function of L and depletion 

width WD. We show that if L is sufficiently small, increasing WD can actually decrease the QE. To determine how long 

L needs to be in a practical GaInNAs junction, we calculate from the QE the short-circuit current density as a function 

of WD and L. This allows us to estimate that Lambipolar needs to be greater than roughly 1 μm in order to obtain enough 

photocurrent for the 4-junction application, giving guidance to the experimental effort to develop such cells. 

15. SUBJECT TERMS

PV; depletion-region collection; solar cells; quantum efficiency; drift field; photocurrent; 

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